A company allocates its annual budget of $260,000 between two departments: marketing and engineering. The marketing department's budget is $10,000...

1. TRANSLATE the problem information

• Given information:
  • Total budget: \(\$260,000\)
  • Marketing budget is \(\$10,000\) less than twice the engineering budget
  • Need to find: marketing department's budget

• What this tells us: We have two unknown budgets with two relationships between them.

2. INFER the approach

• This is a system of equations problem because we have two unknowns and two relationships
• Let M = marketing budget, E = engineering budget
• We'll use substitution since one equation gives us M in terms of E

3. TRANSLATE relationships into equations

• Total budget equation: \(\mathrm{M + E = 260,000}\)
• Budget relationship: "marketing is \(\$10,000\) less than twice engineering" means \(\mathrm{M = 2E - 10,000}\)

4. SIMPLIFY using substitution

• Substitute the second equation into the first:
  \(\mathrm{(2E - 10,000) + E = 260,000}\)
• Combine like terms:
  \(\mathrm{3E - 10,000 = 260,000}\)
• Add 10,000 to both sides:
  \(\mathrm{3E = 270,000}\)
• Divide by 3:
  \(\mathrm{E = 90,000}\)

5. SIMPLIFY to find marketing budget

• Use \(\mathrm{M = 2E - 10,000}\)
• \(\mathrm{M = 2(90,000) - 10,000}\)
  \(\mathrm{M = 180,000 - 10,000}\)
  \(\mathrm{M = 170,000}\)

Answer: C. 170,000

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE skill: Students misinterpret "marketing is \(\$10,000\) less than twice engineering" and write \(\mathrm{M = 2E + 10,000}\) instead of \(\mathrm{M = 2E - 10,000}\).

This leads to the wrong system: \(\mathrm{(2E + 10,000) + E = 260,000}\), giving \(\mathrm{3E = 250,000}\) and \(\mathrm{E ≈ 83,333}\). The resulting marketing budget doesn't match any clean answer choice, causing confusion and guessing.

Second Most Common Error:

Incomplete solution: Students correctly solve for \(\mathrm{E = 90,000}\) but then select this as their final answer, forgetting that the question asks for the marketing budget, not the engineering budget.

This leads them to select Choice A (90,000).

The Bottom Line:

This problem tests whether students can carefully translate word relationships into algebraic expressions and follow through with a complete solution to answer the actual question being asked.